The critical multitype branching process in random environment

Abstract

Consider a population of particles of k different types. Each particle is capable of producing new particles of the different types by branching. The branching process is developing in a random environment (RE) that changes stochastically from generation to generation. This change in the environment affects the probability distribution of the reproduction of the different particles. We let N m have random components equal to the number of particles in the mth generation of each type 1 to k, respectively. Also, let Λ be the first moment matrix for the N m process in RE. Assume that Λ is positively regular. It follows that Λ has an eigenvalue ϱ ∗ of maximum modulus and corresponding right eigenvector v. In a recent article [1], the authors proved that when ϱ ∗ = 1, N m v converges to a random variable W̄ with probability one, under a further restriction on Λ. This result was used by the authors to show that for the ordinary k-type branching process, with no random environment, the Spitzer–Joffe [5] result of convergence to zero in probability of the population could be extended to almost sure convergence. In this article, we give a sufficient condition for the almost sure extinction of the critical k-type branching process in RE. We make special note that when our model is collapsed to the ordinary k-type process, the restriction imposed on the matrix Λ by Spitzer and Joffe is no longer required, thus providing a condition for extinction for the entire class of positively regular matrices with ϱ ∗ = 1.